The flow of corporate control in the global ownership network

We propose a model and algorithm to measure the amount of influence a shareholder has over the flow of corporate control held by the ultimate owners. Existing models of corporate ownership and control either focus on the ultimate owners’ influence or inadequately evaluate the influence possessed by intermediate shareholders in a ownership network. As it extends Network Power Index (NPI) that describes the the power of corporate control possessed by the ultimate owners, our new model, Network Power Flow (NPF), delineates the distribution of ownership influence among shareholders across the network and identifies the channels through which the ultimate owners’ corporate control travel through the global shareholding network. Our analysis of NPI and NPF values for 7 million ultimate owners and 16 million shareholders reveals a new landscape of ownership and control in the global shareholding network that remained opaque before.

of quotas, q j ∈ [1/2, 1], per company j's decision rule, and v is a vector of the value of companies j (e.g., the sales or the number of employees).
Following the Shapley-Shubik power index, we let N j = {i ∈ N | x ij > 0} denote a set of shareholders holding ownership in company j and S j ⊆ N j denote a coalition of shareholders (i.e., a permutation of i ∈ N j ). If the sum of voting rights amassed in a coalition exceeds the quota of company j, i.e., i∈Sj x ij > q j , this coalition is called a winning coalition in j and denoted by W j . If shareholder i belongs to a winning coalition and its withdrawal breaks this coalition, i.e., if S j ∈ W j and S j \{i} ̸ ∈ W j , then shareholder i is a pivot in S j who, as Shapley and Shubik argue, controls decision-making in j, and hence establishes a (direct) control linkage denoted by y ij = 1.
Also following Shapley and Shubik, we assume that every coalition S j available in N J may be realized with equal probability. Then, the next definition follows: Definition 1 (The Shapley-Shubik power index). Shareholder i's power of controlling j is given by the probability that i is a pivot in S j such that where s j and n j are the size of S j and N j , respectively.
We have generalized the Shapley-Shubik in a network setting by taking into account various ways in which shareholder i may obtain indirect control linkages in j through the complex network of shareholding [1]. Let y ij ∈ {0, 1} denote the indirect control relation between i and j, whereȳ ij = 1 indicates that shareholder i holds an indirectly control linkage in company j. One way in which i establishes an indirect control linkage in j is by transitively connecting multiple direct control linkages y ij . More specifically, suppose that shareholder i does not have a direct control linkage to j (i.e., y ij = 0) but it does to k (i.e., y ik = 1). If shareholder k has a direct control linkage to j (i.e., y kj = 1), then i can establish an indirect control linkage to j (i.e.,ȳ kj = 1), by sequencing its voting power from I to k and then to j, using k as an intermediary. LettingȲ j denote a set of linkages that has indirect control j, we make the following assumption about behavior of a winning coalition that controls j: Assumption 1 (Union formation). If shareholders k are under the control of ultimate owner i (i.e.,ȳ ik = 1) and have ownership in j (i.e., x kj > 0), then these shareholders form a union U i j (Ȳ j ) ⊆ N j as a permutation of the entities k.
shareholders A and D over company E, there are two possible ways in which A and D can form a union, {A, D} and {D, A}. That is the order in which a shareholder joins a union matters since per the Shapley-Shubik power index, the power to control a union is given to a pivotal voter that address the question of who is the last piece of the elements in a union for the union to form the winning (majority) coalition?
Moreover, for every union that controls j, we maintain Assumption 1 about voting coordination. Then the next corollary follows from Assumption 1.
Corollary 1 (Integrity of Unions). Every union must maintain integrity so that any member of an union cannot defect from its union to join another in order to form a winning coalition.
That is, any shareholder is prohibited from breaking its union to join another union in order to be a pivot in a new winning coalition. Every union must maintain integrity and any member of an union cannot defect from its union to join another in order to form a winning coalition. Since any union cannot cut across multiple winning coalitions in j, a family of winning coalitions must subsume the entirety of any given union within themselves With these assumptions about voting behavior and the type of winning coalitions being permitted, we now define an individual NPI of i against j in a network G = (N, X, q, v) as the probability that ultimate owner i forms an indirect control linkage to j: where An important and useful characteristic of NPI is that the sum of individual NPI values among all the ultimate owner for each target company j is equal to one (1). As we show in Proposition 1 below in this Appendix, this is guaranteed by the claim that individual NPI is Pareto optimal.
While individual NPIp ij (G) measures the extent to which an ultimate owner has the power to influence the managerial decision-making in a specific target company j, aggregated NPI sums up all the individual NPI values possessed by i in the network and measures its overall power to influence all the companies. We call this quantity aggregate NPI which is given byp i (G) ≡ jp ij (G). As for interpretation, since ultimate owner i's individual NPI with respect to j gives the probability that i controls j, aggregate NPI, that is, the sum of all individual NPIs for i gives the expected number of companies this ultimate owner may indirectly control in the entire network.
We can also weight individual NPI as well as aggregate NPI by the value of a target company. As a corollary, the sum of NPIs weighted by the value of a target company, v j , is the expected value under its control. We call this quantity an aggregate NPI, denoted byp i (G) = j v jpij (G). Now that we have defined NPI, we are ready to define NPF, that is the power of intermediate shareholder in the ownership network. The intermediate shareholder's power is defined as the power of being the last piece in a union that makes the union become a winning coalition over a target company, which is led by its ultimate owner.
That is, an intermediate shareholder's the power is derived from the fact that the shareholder is indispensable for its ultimate owner to establish a control path (i.e., a winning coalition) over a target company.
NPF for k can be interpreted as the expected number of companies in the downstream below k on its control paths. Suppose that we obtain a set of direct control paths (N, Y ) by repeatedly applying the procedures above. An element in the direct control matrix, y kj , is one if k is a pivot in S j with unions and zero otherwise. Then, given this network, the expected number of companies in the downstream of a control path, If an ownership structure entails cross-ownership causing a loop in the ownership network, we instead use 1) is a damping factor to avoid the indeterminacy. Finally, since a direct control network is stochastic, we obtain an aggregated NPF value for company k by taking the average: Note that if there is cross-ownership and hence no loop in the ownership structure in the network, the aggregate NPF value is equivalent to its corresponding aggregate NPI value for every ultimate owner (Proposition 2).

Properties of NPF
In this section, we derive the properties of NPF in the case of no loops in G, i.e., d = 1. Let Y denote a direct control matrix with unions, where y kj is one if player k is a pivot in j with unions U i j (Ȳ j ) and zero otherwise. Similarly, letȲ denote an indirect control matrix, whereȳ ij is one if i establishes an indirect control of j and zero otherwise. Lemma 1. Y n is a matrix such that its ij-th element is one if player i can reach j in n steps and zero otherwise.
Proof. Since Y is an adjacency matrix, its n-th power, Y n , indicates the number of paths between i and j with the length of n. By the definition of a pivot, there exists only one pivot for each company j; if y kj = 1, then for k such that y kj = 1. Since y (1) ij ∈ {0, 1} holds, y (1) ij ∈ {0, 1}, ∀n also holds by induction, implying that there exists only one path between i and j in n links if it exists.
This lemma implies that we can obtain the indirect control matrixȲ as (I − Y ) −1 .

Lemma 2. An indirect controlȲ is equivalent to
Proof. Following the Neumann series, we obtain ( Because we assume that there is no loop, there exists n such that (I − Y ) −1 = n n=0 Y n . Following the lemma 1, (I − Y ) −1 is a reachability matrix whose element indicates whether any two players are connected or not in (N, Y ). This is an indirect control matrix by definition.
The Pareto optimality of an individual NPI and the equivalence of NPI and NPF follow this lemma.

Proposition 2 (Equivalence of NPI and NPF). For ultimate owner i, p i (G) =p i (G).
Proof. From the lemma 2, we obtain p(Y, v) =Ȳ v, implying that p i (Y, v) is equal to jȳ ij v j . By taking an expectation, we obtain the NPF for i, i.e., the aggregate NPI.
The proposition below ensures that the algorithm in the main text consistently estimates NPFs.

Proposition 3 (Consistency). The estimator of NPF,p(G), converges in probability to the true value, p(G), as the number iteration T increases, i.e.,p(G) → p p(G).
Proof. The ij-the element ofp is following the lemma 2. And, as shown in the proof of proposition 2, we have p i (G) = j P (ȳ ij = 1)v j . These mean if T t=1ȳ (t) ij /T converges to P (ȳ ij = 1), then the argument is true. We will show T t=1ȳ For an indirect control structureȲ j , let T (Ȳ j ) be a set of t such thatȲ ij /T (Ȳ j ) and that ofȲ j is T (Ȳ j )/T , the frequency ofȳ ij = 1 can be decomposed as follows T t=1ȳ · T (Ȳ j ) T (2) .
(6) Therefore, if the first part and the second part converge to P (ȳ ij = 1 |Ȳ j ) and P (Ȳ j ), then we obtain the proposition.

Part (1)
We further decompose the first part of Eq. 6 as follows; GivenȲ j , the event of {y 1}} is a series of T (Ȳ j ) times Bernoulli trial with the probability of P (y kj = 1 | W j (Ȳ j )). Follwing [2], Hoeffding's inequality implies which ensures Hence, we show Part (2) Let N l be a set of players j such that the distance of the longest path from any ultimate owner to j is l.
That is, N 0 is the set of ultimate owners and N 1 is the set of copmanies whose shares are owned by only ultimate owners. For k ∈ N 1 , indirect control linkageȳ ik is equal to direct control linkage y ik .
Then, consider company j ∈ N 2 , which are likned with ultimate owners and companies k ∈ N 1 . Indirect control structure above j,Ȳ j , is control linkages between i and k. Therefore, the probability of a specific control sturucture is P (Ȳ j ) = k P (ȳ ik = 1). On the other hand, the empirical distribution ofȲ j is This stochastically converges to k P (ȳ ik = 1) because ultimate owner i being pivot in k is independent for k ∈ N l and the probability of being a pivot is consistently estimated as discussed in part (1).
The empirical distribution T (Ȳ j ′ )/T converges in probability to this quantity because P (ȳ ij = 1 |Ȳ j ) and P (Ȳ j ) are also consistently estimated. The fact the probability of indirect control structureȲ j ′ can be consistently estimated for j ′ ∈ N 3 ensures the probability ofȲ j for j ∈ N 4 is also possible to consistently estimate and so forth. Hence, by induction, we can show for j ∈ N l , ∀l Finally, since T increases T (Ȳ j ), we have T t=1ȳ orp(G) → p p(G).